To put the molecular gas masses in global context, [1]} computed the cosmic molecular gas density using the cosmic-volume-averaged C\(^+\)  luminosity density at \(z\)\(\sim \) 5. For the computation of the latter, the derived C\(^+\)  LF [1]} was integrated down to \(7\times 10^7\,{\rm L_\odot }\) , yielding \(L_{{\rm C^{+}}}\)\(\sim \)\(1.2 \times 10^6\,{\rm L_\odot }\) . Several previous studies [3]}, [4]} and the study based on the ALPINE sample by (see below; [5]}) have derived a linear ratio \(\rm M_{molgas}/L_{{\rm C^{+}}}~=~\text{15--60}\,M_{\odot }/L_\odot \) . Using this ratio, [1]} derived the molecular gas mass density \(\rho _{\rm molgas}~=~\text{(2--7)}\times 10^7\,{\rm M_\odot \,pc^{-3}}\)  at \(z \sim \)  4–6. At a \(2\sigma \)  and 95% confidence level, the estimated \(\rho _{\rm molgas} = \text{(1--14)}\times 10^7\,M_\odot \,{\rm Mpc^{-3}}\) . The evolution of the cosmic molecular gas mass density (left panel in Figure REF ) closely follows the evolution of the cosmic SFR density (see right panel of Figure REF ).
Individual molecular gas masses of the ALPINE galaxies are calculated in different ways using the abovementioned ingredients. [5]} show promisingly that molecular gas masses derived from the RJ dust continuum, C\(^+\)  emission, and dynamical mass agree well with each other within the measurement uncertainties (approx. a factor of two).
The RJ emission at rest-frame \(850\,{\rm {m}}\)  was calculated using a far-IR SED template with a temperature of \(\sim \)\(40\,{\rm K}\)  normalized to the ALPINE/ALMA measurement at rest-frame \(150\,{\rm {m}}\)  (see Section REF  for more details).
The dynamical mass was calculated for a subset of 18 nonmerger galaxies classified by [8]} with C\(^+\)  emission at S/N\(>5\)  and with C\(^+\)  halo size measurement from [9]}.
Figure REF  shows the gas fraction and depletion times at \(z=\)  4–6 derived by [5]} in comparison to measurement from CO emission at lower redshifts from the literature [11]}, [12]}, [4]}.
Overall, the rise of the molecular gas fraction traces the evolution of the sSFR (e.g., [14]}, [15]}); it increases with redshift up to \(z\)\(\sim \) 3–4 (reaching a value of \(\sim \)\(63\%\)  at \(z\)\(\sim \) 5) and then flattens out at earlier cosmic times.
Individual measurement of gas masses for ALPINE galaxies show a large scatter, which is likely the physical reason for the large range in the measurement of the cosmic molecular gas mass density at \(z\)\(\sim \) 5 (indicated by the vertical size of the boxes in Figure REF , left panel).
Furthermore, the study finds a continuous decline in the molecular gas depletion time scale (\(t_{\rm depl} = M_{\rm gas}/{\rm SFR}\) )with the total SFR derived from the infrared and UV SFRs. of a factor of 2–3 from \(z=0\)  to \(z=6\) , suggesting a mild enhancement of star-formation efficiency towards high redshifts. This trend is in agreement with the extrapolation of the parametrization by [16]} based on \(z<4\)  as well as [17]}, but significantly shallower than \((1+z)^{-1.5}\) , which is predicted by semianalytical and cosmological simulations based on the framework of the bathtub model (e.g., [18]}, [19]}, [20]}, [21]}). [5]} also show that the depletion time depends on the sSFR (which has been shown to be true also at lower redshifts, e.g., [11]}), which may be the reason for the change of depletion time across cosmic time. Still, compared to the upsizing of the gas reservoirs, the increase in star-formation efficiency seems to play a minor role in reaching the high average SFRs at high redshifts, in agreement with the study by [16]}.
